Financial Econometrics

Copulas

Semester II 2016

Introduction

\section{Introduction}

GIF

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References

Goal

Fields where copulas are applied

\section{Understanding Copulas}

Introduction to copulas

Introduction to copulas (Sklar)

Introduction to copulas (Sklar cont.)

Lets have a look visually

Definitions and basic properties

Definitions and basic properties (cont)

Definition (Copula): A d-dimensional copula is the distrubutiom function \(\mathcal{C}\) of a random vector \(U\) whose components \(U_k\) are uniformly distributed \begin{align} C(u_1, \ldots, u_d) = P(U_1 \leq u_1, \ldots, U_d \leq u_d), (u_1, \ldots, u_d) \in (0,1)^d \end{align}\vspace{-1.5cm} Thus Sklar’s theorem states: \begin{align} C(F_1(x_1), \ldots,F_d(x_d)) &= P(U_1 \leq F_1(x_1), \ldots, U_d \leq F_d(x_d))\notag\\ &= P(F_1^{-1}(U_1) \leq x_1, \ldots, F_d^{-1}(U_d) \leq x_d)\notag\\ &= F(x_1, \ldots, x_d) \end{align}

Joint distribution function:

Kendall’s Tau

\section{Copulas}

Meeting the generator

The Gaussian copula \(C_p^{Ga}\) is a distribution over the unit cube \([0,1]^{d}\). This represents a linear correlation matrix \(\rho\). \(C_p^{Ga}\) is defined as the distribution function of a random vector \((\psi(X_1),\ldots, \psi(X_d))\), where \(\psi\) is the univariate standard normal distribution function where \(X \sim, N_d(0,\rho)\) \begin{align} C_p^{Ga}=\Phi _{R}\left(\Phi ^{-1}(u_{1}),\dots ,\Phi ^{-1}(u_{d})\right) \end{align}\vspace{-1.5cm}

where \(\Phi^{-1}\) is the inverse cumulative distribution function of a standard normal and \(\Phi_{R}\) is the joint cumulative distribution

Student T Copula

The student’s t-copula \(C_{\upsilon,\rho}^t\) of \(d\)-dimension can be characterised by parameter \(\upsilon\geq0\) degrees of freedom and linear correlation matrix \(\rho\). The random vector \(X\) has a \(t^d(0,\rho\upsilon)\) distribution with univariate function \begin{align} C_{\upsilon,\rho}^t &= \mathcal{P}(t_\upsilon(X_1)\leq(u_1), \ldots, t_\upsilon(X_d)\leq u_d)\\ &=t_{\upsilon,\rho}^d(t_{\upsilon}^{-1}(u_1),\ldots, t_\upsilon^{-1}(u_d)) \end{align}

Archimedean Copulas

Illustration

Some families

Some families

See Joe (2014)

Vine-Copulas

Under suitable differentiability conditions, any multivariate density \(F_{1\ldots n}\) on \(n\) variables may be represented in closed form as a product of univariate densities and (conditional) copula densities on any R-vine \(V\)

Vine copulas kurowicka2006

The R-vine copula density is uniquely identified according to Theorom 4.2 of @kurowicka2006: \begin{align} c(F_1(x_1),\cdots, F_d(x_d)) = \prod_{i=1}^{d-1}\prod_{e\in E_i}c_{j(e),k(e)|D(e)}\left(F(x_{j(e)}|\mathbf{x}_{D(e)})\right) \label{r_vine_eq} \end{align}

Introduction of Copula-MGARCH model

References

Carmona, René. 2014. Statistical Analysis of Financial Data in R. Vol. 2. Springer.

Charpentier, Arthur. 2014. Computational Actuarial Science with R. CRC Press.

Joe, Harry. 2014. Dependence Modeling with Copulas. CRC Press.

Nelsen, Roger B. 2007. An Introduction to Copulas. Springer Science & Business Media.

Ruppert, David. 2011. Statistics and Data Analysis for Financial Engineering. Springer.